# "A statistical definition of value": a critical comment.

In Max Kummerow's above referenced Journal article (October 2002), he argues against traditional verbal definitions of market value and instead favors a statistical definition of value. Some of the issues related to "thinking statistically" about definitions and valuations were analyzed in earlier works by Kummerow (1) and Colwell. (2) There are a number of important points in these articles, especially in Kummerow's, but it will be argued here that he has drawn too far-reaching conclusions in his most recent 2002 article.Main arguments and conclusions in Kummerow (2000) (3)

The best definition of market value is a simple Ratcliff-type definition (4) where market value is defined as the most probable price given that the property is marketed in the normal way. Nothing should be said about knowledgeable actors; in valuation we look at the market as it is. Nothing needs to be said about willing buyers and sellers; we ask a hypothetical question about what would happen if the property were put on the market and marketed in the normal way--only actors who are willingly interested in buying will be met.

When a property is sold, the observed price is one of a number of possible prices. There is a certain distribution of possible prices, and it is important to discuss how this distribution looks. Is it "peaked" around a certain price, as it would be if there were a large market of rather homogenous properties and a considerable number of similar actors on the market? Is the distribution rather "flat," as it might be if the property were special and if the actors had different preferences and bargaining skills? Understanding that there is such a distribution of possible prices is important, for example, when judging whether a certain appraisal is good or bad, or whether a negligence claim is justified.

Even if the distribution of possible prices is important Kummerow states, "Of course, specific value figures are needed for most of the purposes that lead to valuations ... But value is added for most clients if the appraiser also reports on the possible variation of prices from this point estimate." He discusses the kind of information that can be used as "clues to the unobservable price distribution." He also has an important discussion about error trade-offs, but there is no reason to go deeper into these issues here.

Points of dispute with Kummerow (2002)

A critique of verbal definitions

Kummerow makes a distinction between traditional verbal definitions and a statistical definition of value. He starts by saying, "Over the past decades, courts and professional societies have struggled to clarify the meaning of 'value' by changing the verbiage in value definitions." In response he believes that "it is time to leave the verbiage behind and switch to a statistical definition instead."

Kummerow's description of the alternatives, however, is misleading. Look at a simple version of a standard definition: The market value is the most probable price if the property is marketed in a normal way. Next look at a simple version of my interpretation of Kummerow's alternative definition: To give information about the market value is the same as estimating the parameters of the subject property's possible price distribution. To me both of these definitions are verbal. They use words to state the content of the definition. Both state the goal when estimating the market value of a property. The question is not whether a definition is verbal or not, but whether it is precise enough and whether it creates a concept that is useful.

Should a definition focus on a specific number or on a distribution of possible prices?

The most important difference between any of the traditional definitions of market value and the definition Kummerow proposes is that the traditional definitions all focus on a specific number. The market value is the expected or most probable price, given certain circumstances. Kummerow's statistical definition focuses instead on a price distribution.

But as he himself argues in his earlier article there are a number of situations where specific value figures are needed, e.g., in the accounts of a firm or as information added to a balance sheet. Tax purposes are another obvious situation. The definition also needs to be standardized to fill its role. Kummerow writes that a proposed statistical definition of value should include: "Statements of assumptions about the circumstances of sale that may influence the possible price distribution including legal rights valued, date of sale, method of sale, time on the market.... "

Kummerow is of course correct in saying that the expected price and the distribution of possible prices for a property will depend on, for example, the method of sale and the time on the market. The standard definition, however, provides a specific set of circumstances (the "normal case"). Such standardization is important if information about market value is to be easily communicated and comparable.

Actors on a market might be interested in the expected price given circumstances other than the one in the standard definition: Will the expected price go up if I choose a non-standard sales method or wait longer to sell? But that people might be interested in this is not a convincing argument against using standardized assumptions in the definition of market value. Such a definition does not say that only the market value so defined is interesting.

Neither is the fact that additional information is important in a number of contexts a convincing argument for changing the definition of value, but it could very well be an argument for changing appraisal practices. Both a buyer and a seller should reasonably be interested not only in the most probable price, but also in the distribution of the possible price and probable price given certain non-standard assumptions.

Are traditional definitions not precise enough compared to a statistical definition?

Standard definitions refer to concepts such as expected price or probable price. The criticism can be made that these terms can be interpreted in several different ways. Do the terms refer to what statisticians call the mean, median or mode? In a statistical definition of value, one of these more precise terms is used.

I can agree that standard definitions usually do not clarify this, but the question is whether this imprecision matters in practice. In most valuations there is not enough information to estimate the distribution of possible prices in more than a rough way And in such a case there is not enough information to estimate any difference between the mean, median and mode. Kummerow also wants the appraiser to estimate measures of skewness and kurtosis. However, in the section on "Error Trade-Offs" in Kummerow (2000), he shows that appraisers' traditional method of focusing on a small number of properties that are most similar to the valuation object can be rational. But if we use only a small number of comparables, it is impossible to estimate any details about the distribution of possible prices.

Colwell makes another point in this context: "It should be recognized that any selling price, including the most probable selling price, is not very probable. Once this is accepted, the appeal of Ratcliff's definition should be diminished greatly." (5) This is not convincing. If an appraiser states that the most probable price of an office complex is $15 million, this must be interpreted as saying that a price in the range of $14.5 to $15.5 million is more probable than a price within the ranges of $13.5 to $14.5 million or $15.5 to $16.5 million. And while there might well be enough information to say that there is a high probability for a price to be in the range of $14.5 to $15.5 million, there is nothing wrong in saying that $15 million is the most probable price.

Kummerow presents the following argument for the greater precision of statistical definitions compared to verbal definitions: "Instead of saying 'well-informed buyers' and then wondering what that means, a statistical valuation simply cuts off outliers above or below three standard deviations from the mean ... This provides a dear and replicable method for operationalizing the concept of 'well-informed' or 'not compelled to sell.'"

Let us assume that we have made a valuation using a Ratcliff-type definition. We have estimated the most probable price given the characteristics of the actual market. Suppose now that someone wants to know the probable price of the property in a situation where the actors are well-informed. I do not think that such a person would be satisfied if we just cut off outliers and said, "Here is the answer." This is not an operationalization of "well-informed," but something completely different. There are a number of ways to make the concept "well-informed" more precise, e.g., by looking at the kind of information experienced actors collect. (6) A second problem with Kummerow's idea on this point is that if we estimate the market value from a small number of similar properties, then the standard deviation cannot be estimated with any kind of precision and, therefore, the cut-off points become rather arbitrary.

Relying on statistical theory to get a more precise definition is also problematic. First, a central concept in Kummerow's definition is "possible price." What does this really mean? It is easy to give an interpretation of "possible" when throwing a six-sided die and thinking about the probability of certain outcomes. But what do we mean by "possible" in the context of property sales? Second, within probability theory there are controversies between those who interpret probability in terms of frequencies (which seems to be the alternative Kummerow has chosen) and those who see probability as a measure of rational degree of belief. (This interpretation is used in Lind 1998). (7) Summing up this section, there is no convincing argument that standard definitions are so vague that they have to be replaced with a completely different type of definition.

Should the valuer present forecasts of the stability of the estimates over the relevant period?

Kummerow writes: "Even more valuable is an idea of the future direction of market values. Forecasts should become a standard part of value definition and appraisal reports because they add a time dimension through a graph of value over time. Market inefficiency and market cycles are partly due to lack of forward looking valuation methods." It could just as well be argued that inefficiencies and cycles occur because actors on the market forget the past! When many were caught up in the "new economy exuberance" a few years ago they looked at the future, but they dramatically overestimated the future profitability of the new-economy firms. At the time, however, there was no way of showing that these predictions were wrong. There was very little on which to base a probability judgment.

Appraisers should guard their role as experts, but no one can be an expert about the future of a specific market. The role of those In business is to use their intuition, guess the future and take the risks. Experts should make statements based on facts and established theories, talking only about the past and the present. One part of the present is, of course, expectations about the future, but appraisers should not claim that they can evaluate these expectations.

Kummerow has an important point in that the risk of "bubbles" might

be reduced if the appraiser added a time dimension, but that addition should perhaps cover the past instead of the future. Knowing how the market has developed over the last 10 to 15 years, for example, might reduce the risk of what Shiller calls irrational exuberance. (8) A proposal that property appraisers include such information about the past value of the property is put forward in Ekelid, et al. (9)

Criticisms of Kummerow's 2002 article should not overshadow a number of very important points he has made in his writings, e.g., that the observed price is only one of the possible prices; that a comparison between observed prices for pairs of properties that differ in only one characteristic gives no certain information about how much this characteristic affects the market value; that it is important to present information about not only the market value, but also about the distribution of possible prices; that there is information that can be used to estimate such a distribution; and that the number of comparables should be chosen after an explicit "error trade-off" evaluation.

In my opinion, however, it would be a mistake to replace the standard type of definitions of market value with the statistical definition proposed by Kummerow.

Hans Lind, PhD

Stockholm, Sweden

Response to "A Statistical Definition of Value": A Critical Comment

Starting with points of agreement, I cannot summarize better than Professor Lind who wrote: "that the observed price is only one of the possible prices; that a comparison between observed prices for pairs of properties that differ in only one characteristic gives no certain information about how much this characteristic affects the market value; that it is important to present information about not only the market value, but also about the distribution of possible prices; that there is information that can be used to estimate such a distribution; that the number of c0mparables should be chosen after an explicit 'error trade-off' evaluation."

These views have practical implications, such as the likelihood that sale prices will not match valuations exactly. Appraisals can differ from sale prices in particular cases due to random sale price variation, but should produce unbiased estimates on average over a large sample of valuations. Look at summary statistics on appraisal errors rather than individual appraisal outcomes to judge the quality of appraisal work.

Also, paired sales analysis should look at more than one paired sale because the difference in prices in a single case could include random errors and, therefore, be misleading. One might attribute a value effect to a swimming pool, when the price difference actually occurred because Jones paid too much for the house with a pool, while Smith got a good deal on the house without a pool. Summary statistics on a sample of 20 paired sales would give a better idea of the expectation and range of values of swimming pools.

In examining comparable sales, it is helpful to think not of "the price" of the property that sold, but to interpret the observed sale price as one out of a range of prices that might have occurred. Variations in indicated values from a sample of comparable sales gives an idea of the variance of the possible price distribution of the subject property. Valuations that estimate variability of prices should be worth more to clients who are concerned with risk and investment outcomes.

Lind and I probably agree more than we disagree. Our disagreement is on two main points: 1) the most useful definition of value and 2) whether valuers should forecast prices. Lind points out that my statistical definition is expressed in words, i.e., "estimating the parameters of the subject property's possible price distribution." Yes, that is verbiage. But it could be rewritten in math symbols as

[SIGMA]wiXi / n

(average of n adjusted, weighted sale prices Xi) and so on for the standard deviation of the indicated values or other summary statistics of the possible price distribution. The words can be translated into math.

However, I did include verbiage similar to that in the Appraisal Institute's 12th edition of The Appraisal of Real Estate in describing the actual conditions of sale. The verbiage does matter and the circumstances of the sale have to be mentioned, except that (like Ratcliff and The Appraisal of Real Estate), I believe the actual circumstances of the sale should be mentioned rather than assumptions about possibly counterfactual perfect market conditions (informed buyers, etc.)

Professor Lind uses the word "operationalize." How do we actually come up with the number reflecting the definition? In chemistry, methods are called "protocols" laying out step-by-step processes used to get results. When the protocols are followed by anyone (first wash the glassware, then heat the material for two hours at 400 degrees, etc.), results should attain a predictable level of accuracy and everyone should get the same answer no matter who does the lab work. Unfortunately we cannot identify steps certain to guarantee a particular level of accuracy in valuations because the data is too heterogeneous. Our "chemicals" are seldom pure.

It seems difficult to operationalize how one would go from a sale price where the buyer was less informed to one where the buyer is more informed. I would like to just accept the transaction price as evidence (unless obviously an outlier like a non-arms-length sale) and not try to adjust for variations from assumed conditions.

It is probably easier to operationalize a statistical value definition using a statistical estimator. An estimator is the calculation used to find an estimate. Adjustment grids are price estimators. Hedonic regression models are price estimators. We can think of whole families of alternative estimators. Different estimators would give different value estimates. If we can state the estimator and protocol precisely (e.g., use a sample size of five comparable sales and adjust $55/sq. ft. for size differences), then everyone would get the same answer, provided we had an unambiguous method at each step.

A lot depends on the data. In Western Australia the Torrens title system provides us with a complete list of sales. So if I can describe an unambiguous protocol for picking comparable sales and adjusting them, etc. the method should produce an unbiased estimate. Agreement between appraisers would not be as likely, I think, under the traditional value definitions.

It does require a leap of faith to assume that performing statistical analyses on a set of sales data will beat traditional valuation methods based on subjective judgments. There is never enough data to completely replace an appraiser's judgment; experience and subjective interpretations are usually needed. That is what Pace, Sirmans and Slawson (1) pointed out when they said that "statistically challenged" appraisers usually produce more accurate value estimates than PhDs with computers at universities. But their article then goes on to perform a "near neighbors" spatial statistics modeling exercise that does produce precise estimates.

Lind points out that as the sample size gets smaller (and, therefore, more homogeneous and less prone to adjustment errors), the statistical estimates become more prone to random variation (i.e., less valid and precise) because of small sample size. Whereas, if we use a large enough sample to allow precise, stable estimates, then the properties may be so diverse that adjustment errors become large. This is the error trade-off dilemma.

Whether there are enough sales of similar properties and enough data on the characteristics of those properties to allow statistical protocols to produce accurate estimates is a question to which there is no general answer. It depends on the data. But as data improves, appraisals can become more objective and require less guesswork An important point is that any valuation method faces this same problem. The experienced valuer relying on experience rather than statistics will also have to make interpretations with limited evidence.

Finally, the forecasting issue. The allegation is that appraisers cannot be trusted to forecast because (a) they do not know how to forecast, lacking specialized forecasting expertise, and (b) forecasting leaves room for making arbitrary and biased estimates (rather than using only current market evidence) that may mislead markets. I reply that it is the basic religion of finance theory that all values are forward looking--the amount paid for assets depends on expected future benefits of ownership (cash flows, enjoyment of occupancy or other benefits). If valuers are not competent to forecast property markets, who is?

Certainly appraisers who take forecasting seriously might need to study econometrics with Bill Wheaton at MIT to improve their skills, or hire a new generation of valuers with more training in forecasting. But even those with no more than local market knowledge and common sense may have a better idea than their clients about where the market is headed. As I discovered by purchasing several finance books with "valuation" in their titles, in stock and bond markets, the term "valuation" means forecasting future values (as in identifying undervalued assets) and this adds far more value for clients than simply reporting the current price. Of course, dishonest analysts kept a "strong buy" recommendation on Enron as it fell from $95 to 20 cents, so it is easy to see where we get our fear of forecasting.

Another point, however, is that it is possible for appraisers to mislead clients by not forecasting. In Australia "never mind tomorrow" valuations, when oversupply was obviously pending, led to large professional negligence claims. Appraisers have a duty of care to report their opinions (based on evidence) that might be relevant to their clients' interests or decisions. If it seems likely that the market will correct, clients should be informed.

The cautions expressed by Lind are valid concerns, but if they are kept in mind and forecasting methods not abused, database technology and statistical methods might prove useful in making valuations more objective and appraisal services more valuable.

Max Kummerow, PhD

Perth, Western Australia

(1.) Max Kummerow, "Thinking Statistically About Valuations," The Appraisal Journal (July 2000): 318-326.

(2.) Peter F. Colwell, "A Statistically Oriented Definition of Market Value," The Appraisal Journal (January 1979): 53-59.

(3.) Similar conclusions are reached in Hans Lind, "The Definition of Market Value. Criteria for Judging Proposed Definitions and an Analysis of Three Controversial Components," Journal of Property Valuation and Investment 16, no. 2 (1998): 159-175.

(4.) Richard U. Ratcliff, Valuation for Real Estate Decisions (Santa Cruz, CA: Democrat Press, 1972).

(5.) Colwell, 59.

(6.) This is discussed in more detail in Lind (1998).

(7.) A presentation of the different schools of thought can be found in Ian Hacking, An Introduction of Probability and Inductive Logic (Cambridge: Cambridge University Press, 2001).

(8.) Robert J. Schiller, Irrational Exuberance (Princeton: Princeton University Press, 1999).

(9.) Mats Ekelid, Hans Lind, Stellan Lundstrom and Erik Persson, "Treatment of Uncertainty in Appraisals of Commercial Properties: Some Evidence from Sweden," Journal of Property Valuation and Investment, 16, No. 4 (1998): 386-397.

(1.) R. Kelley Pace, C.F. Sirmans, and V. Carlos Slawson, Jr., "Are Appraisers Statisticians?" Real Estate Valuation Theory (Boston: Kluwer Academic Publishers, 2002), 31-44.

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Title Annotation: | letters to the editor |
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Author: | Lind, Hans |

Publication: | Appraisal Journal |

Date: | Jul 1, 2003 |

Words: | 3718 |

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